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(Please write your Roll No. immediately)

Roll No. ..................................

End-Term Examination

First Semester [B.Tech] – December 2003

Paper Code : ETMA – 101 Subject : Applied Mathematics - I

Time : 3 Hours

Maximum Marks : 75

Note : Attempt any 5 questions in all. At least two questions from each section. All questions carry equal marks.

SECTION-A

Q1

( a )

State and prove Lagrange's Mean value theorem and verify it for the function f(x) = sin x + cos x in [0,π/2].

5

( b )

If y = (sin-1 x)2, prove that (1 – x2) y2 - xy1 = 2 and also show that
(1 – x2) yn+2 – (2n + 1) xyn+1 – n2yn = 0

5

( c )

If ρ1 and ρ2 are the radii of curvature at the extremities of two conjugate diameters of the ellipse
x2/a2 + y2/b2 = 1, then prove that (ρ12/3 + ρ22/3)(ab)2/3 = a2 + b2

5

Q2

( a )

Use Taylor's theorem to evaluate √10 correct to four significant figures.

5

( b )

Evaluate:

π<BR>∫(x dx)/(a2 cos2 x + b2 sin2 x)2<BR> 0

5

( c )

Determine the asymptotes of 4x3 - 3xy2 - y3 + 2x2 - xy - y2 - 1 = 0

5

Q3

( a )

Show that
(i)

gamma(3/2-x)gamma(3/2+x) = (1/4 + x) pisec (pi x) provided -1<2x<1
(ii)

0.inf?cos(bz1/n) dz = gamma ( (n + 1) cos(  n p) ) / bn

5

( b )

Test for the convergence of the following series : -

  1. x2 + (22 / 3.4) x4 + (22 .42 / 3.4.5.6) x6 + (22 .42 .62 / 3.4.5.6.7.8) x8 + .....

  2. log(2/1) – log(3/2) + log(4/3) – log(5/4) +.....

5

( c )

The velocity v(km/min) of a moped, which starts from rest is given at fixed intervals of time. Estimate approximately the distance covered in 20 minutes by applying Simpson's 1/3 rule.

5


t

2

4

6

8

10

12

14

16

18

20



v

10

18

25

29

32

20

11

5

2

0


Q4

( a )

Determine the length of the loop of the curve 9y2 = (x-3)(x-6)2

5

( b )

Find the area common to the cardiode r = a(1+cos θ) and the circle r = (3/2)a and also the area of the remainder of the cardiode.

5

( c )

If the curve r = a+b cos θ (a>b) revolves about the initial line, show that the volume generated is (4/3) π a (a2 +b2 )

5

SECTION-B

Q5

( a )

If v = (x2 +y2 +z2 )-1/2 show that



5

( b )

If


prove that:
(i)





(ii)



5

( c )

Expand sin(xy) in powers of (x-1) and (y-π/2) up to and including second degree terms.

5

Q6

( a )

Find the volume of the tetrahedron bounded by the coordinate planes and the plane x + y + z = 1

5

( b )

Sketch the region of integration and evaluate:




5

( c )

Changing into polar coordinates and hence evaluate:




5

Q7

( a )

Solve:
( i )


( ii )

5

( b )

Solve : ( 2xy cos x2 – 2xy +1 )dx + ( sin2 - x2 )dy = 0

5

( c )

Solve :




5

Q8

( a )

If u1=(x2 x3)/x1 , u2=(x1 x3)/x2 , u3=(x1 x2)/x3 , then evaluate J ( u1, u2, u3).

5

( b )

Solve :



5

( c )

Solve :



5

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